Brussels-London XXII

Higgs bundles

University College London, 23rd March 2023

After a two year hiatus, the Brussels-London seminar returns!

The seminar will take place at University College London. The morning session and lunch will take place in 124 Gideon Schreier Lecture Theatre, Bentham House. The afternoon talks will take place in XLG1 Chemistry Lecture Theatre, Christopher Ingold Building. The afternoon tea/coffee break will be in room 502, Mathematics, 25 Gordon Street.

To attend this event, you should register in advance.

Laura Schaposnik. A map of Higgs Bundles, Generalized Hyperpolygons and More.

In this talk we will introduce Higgs bundles and generalized hyperpolygons, and look into different directions that have attracted attention within the area in the last decade. In particular, we shall see how Mirror Symmetry can be seen in terms of branes within the Hitchin fibration and show that, under certain assumptions on flag types for generalized hyperpolygons, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Time permitting, we will also see how I have been using my geometric background to answer questions in other areas of science. Much of the talk follows recent work joint with Steven Rayan.

Laura Fredrickson. Hyperkahler Metrics Near Semi-Flat Limits.

We make rigorous (a generalization of) the formalism of Gaiotto, Moore, and Neitzke for constructing hyperkahler manifolds near semi-flat limits. In particular, we account for the effects of complicated (e.g., densely wall crossing) BPS spectra and singular fibers. This provides a general framework for proving results about Gromov-Hausdorff collapse of hyperkahler manifolds to semi-flat limits and completes the Strominger-Yau-Zaslow conception of mirror symmetry for hyperkahler manifolds at the level of hyperkahler geometry (as opposed to only constructing one complex structure). We characterize the dependence of the periods of the three canonical Kahler forms on the natural parameters of the construction, and in particular prove that for non-compact manifolds this dependence is affine-linear. Specializing to the case of moduli spaces of weakly parabolic SU(2) Higgs bundles on a sphere with four punctures, we prove that this construction yields all such manifolds which are sufficiently close to the semi-flat limit. This talk is based on joint work with Arnav Tripathy and Max Zimet.

Nigel Hitchin. The odd integrable system.

The completely integrable system on the moduli space of Higgs bundles is based on evaluating on a Higgs field a basis for the invariant polynomials on the Lie algebra. If we take invariant alternating forms instead of symmetric ones then there is another story. On the one hand it can be interpreted in supermanifold terms as an integrable system for an odd symplectic structure but a more promising viewpoint is its relationship with the Hochschild cohomology of the moduli space of stable bundles.